| Step |
Hyp |
Ref |
Expression |
| 1 |
|
clscld.1 |
|- X = U. J |
| 2 |
|
sscon |
|- ( T C_ S -> ( X \ S ) C_ ( X \ T ) ) |
| 3 |
2
|
adantl |
|- ( ( S C_ X /\ T C_ S ) -> ( X \ S ) C_ ( X \ T ) ) |
| 4 |
|
difss |
|- ( X \ T ) C_ X |
| 5 |
3 4
|
jctil |
|- ( ( S C_ X /\ T C_ S ) -> ( ( X \ T ) C_ X /\ ( X \ S ) C_ ( X \ T ) ) ) |
| 6 |
1
|
clsss |
|- ( ( J e. Top /\ ( X \ T ) C_ X /\ ( X \ S ) C_ ( X \ T ) ) -> ( ( cls ` J ) ` ( X \ S ) ) C_ ( ( cls ` J ) ` ( X \ T ) ) ) |
| 7 |
6
|
3expb |
|- ( ( J e. Top /\ ( ( X \ T ) C_ X /\ ( X \ S ) C_ ( X \ T ) ) ) -> ( ( cls ` J ) ` ( X \ S ) ) C_ ( ( cls ` J ) ` ( X \ T ) ) ) |
| 8 |
5 7
|
sylan2 |
|- ( ( J e. Top /\ ( S C_ X /\ T C_ S ) ) -> ( ( cls ` J ) ` ( X \ S ) ) C_ ( ( cls ` J ) ` ( X \ T ) ) ) |
| 9 |
8
|
sscond |
|- ( ( J e. Top /\ ( S C_ X /\ T C_ S ) ) -> ( X \ ( ( cls ` J ) ` ( X \ T ) ) ) C_ ( X \ ( ( cls ` J ) ` ( X \ S ) ) ) ) |
| 10 |
|
sstr2 |
|- ( T C_ S -> ( S C_ X -> T C_ X ) ) |
| 11 |
10
|
impcom |
|- ( ( S C_ X /\ T C_ S ) -> T C_ X ) |
| 12 |
1
|
ntrval2 |
|- ( ( J e. Top /\ T C_ X ) -> ( ( int ` J ) ` T ) = ( X \ ( ( cls ` J ) ` ( X \ T ) ) ) ) |
| 13 |
11 12
|
sylan2 |
|- ( ( J e. Top /\ ( S C_ X /\ T C_ S ) ) -> ( ( int ` J ) ` T ) = ( X \ ( ( cls ` J ) ` ( X \ T ) ) ) ) |
| 14 |
1
|
ntrval2 |
|- ( ( J e. Top /\ S C_ X ) -> ( ( int ` J ) ` S ) = ( X \ ( ( cls ` J ) ` ( X \ S ) ) ) ) |
| 15 |
14
|
adantrr |
|- ( ( J e. Top /\ ( S C_ X /\ T C_ S ) ) -> ( ( int ` J ) ` S ) = ( X \ ( ( cls ` J ) ` ( X \ S ) ) ) ) |
| 16 |
9 13 15
|
3sstr4d |
|- ( ( J e. Top /\ ( S C_ X /\ T C_ S ) ) -> ( ( int ` J ) ` T ) C_ ( ( int ` J ) ` S ) ) |
| 17 |
16
|
3impb |
|- ( ( J e. Top /\ S C_ X /\ T C_ S ) -> ( ( int ` J ) ` T ) C_ ( ( int ` J ) ` S ) ) |